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Math. UAB 27 (1983), 81–108. [S1] W. Schlag, A generalization of Bourgain’s circular maximal theorem, J. Amer. Math. Soc. 10 (1997), 103-122. [S2] W. Schlag, A geometric proof of the circular maximal theorem, Duke Math. , to appear. [SS] W. Schlag, C. D. Sogge, Local smoothing estimates related to the circular maximal theorem, Math. Res. Lett. 4 (1997) 1–15. ¨ lin, Regularity of solutions to Schr¨ [Sj] P. Sjo odinger equations, Duke Math. J. 55 (1987), 699–715. M. Stein, Harmonic Analysis, Princeton University Press, 1993.

TAO AND A. VARGAS GAFA References [B] [Bo1] [Bo2] [Bo3] [BoC] [C] [Ch] [Co1] [Co2] [Co3] [DK] [FS] [FoK] [KT] [KlM1] [KlM2] [KlM3] [KlT] M. Beals, Self-Spreading and strength of singularities for solutions to semilinear wave equations, Annals of Math 118 (1983), 187–214. J. Bourgain, A remark on Schrodinger operators, Israel J. Math. 77 (1992), 1–16. J. Bourgain, Estimates for cone multipliers, Operator Theory: Advances and Applications 77 (1995), 41–60. J. M. Stein, Princeton University Press (1995), 83–112.

8 Acknowledgements The first author was partially supported by NSF grant DMS-9706764. The second author was partially supported by the Spanish DGICYT (grant number PB97-0030) and the European Comission via the TMR network (Harmonic Analysis). We thank Sergiu Klainerman, Hart Smith, Andreas Seeger, and Chris Sogge for informing us of some of the applications mentioned above. 256 T. TAO AND A. VARGAS GAFA References [B] [Bo1] [Bo2] [Bo3] [BoC] [C] [Ch] [Co1] [Co2] [Co3] [DK] [FS] [FoK] [KT] [KlM1] [KlM2] [KlM3] [KlT] M.

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